To multiply \( \frac{2}{3} \) by \( 5 \frac{2}{3} \), we first need to convert the mixed number \( 5 \frac{2}{3} \) into an improper fraction.
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Convert \( 5 \frac{2}{3} \) to an improper fraction: \[ 5 \frac{2}{3} = 5 \cdot 3 + 2 = 15 + 2 = 17 \] So, \( 5 \frac{2}{3} = \frac{17}{3} \).
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Now, multiply \( \frac{2}{3} \) by \( \frac{17}{3} \): \[ \frac{2}{3} \times \frac{17}{3} = \frac{2 \times 17}{3 \times 3} = \frac{34}{9} \]
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Next, convert \( \frac{34}{9} \) to a mixed number. To do this, divide the numerator by the denominator: \[ 34 \div 9 = 3 \quad \text{(since } 9 \times 3 = 27 \text{)} \] The remainder is: \[ 34 - 27 = 7 \] Thus, \( \frac{34}{9} = 3 \frac{7}{9} \).
Therefore, the product of \( \frac{2}{3} \) and \( 5 \frac{2}{3} \) is: \[ \boxed{3 \frac{7}{9}} \]